Related papers: On the Classifications of Scalar Evolution Equatio…

The second order partial difference equation of two variables $ \CD u:= A_{1,1}(x) \Delta_1 \nabla_1 u + A_{1,2}(x) \Delta_1 \nabla_2 u + A_{2,1}(x) \Delta_2 \nabla_1 u + A_{2,2}(x) \Delta_2 \nabla_2 u & \qquad \qquad \qquad \qquad + B_1(x)…

In this paper we consider the homogenization problem of nonlinear evolution equations with space-time non-locality, the problems are given by Beltritti and Rossi [JMAA, 2017, 455: 1470-1504]. When the integral kernel $J(x,t;y,s)$ is…

We prove existence and uniqueness results for (mild) solutions to some non-linear parabolic evolution equations with a rough forcing term. Our method of proof relies on a careful exploitation of the interplay between the spatial and time…

Solutions $u(x)$ to the class of inhomogeneous nonlinear ordinary differential equations taking the form \[u'' + u^2 = \alpha f(x) \] for parameter $\alpha$ are studied. The problem is defined on the $x$ line with decay of both the solution…

We propose some finite element schemes to solve a class of fourth-order nonlinear PDEs, which include the vector-valued Landau--Lifshitz--Baryakhtar equation, the Swift--Hohenberg equation, and various Cahn--Hilliard-type equations with…

The paper deals with second order abstract linear partial differential equations (LPDE) over a partial differential field with two commuting differential operators. In terms of usual differential equations the main content can be presented…

We develop the concept and the calculus of anti-self dual (ASD) Lagrangians which seems inherent to many questions in mathematical physics, geometry, and differential equations. They are natural extensions of gradients of convex functions…

We develop a theory connecting the following three areas: (a) the mean field equation (MFE) $\triangle u + e^u = \rho\, \delta_0$, $\rho \in \mathbb R_{>0}$ on flat tori $E_\tau = \mathbb C/(\mathbb Z + \mathbb Z\tau)$, (b) the classical…

Fractional difference equations provide a flexible mathematical framework for modeling complex systems with memory, hereditary, and non-local effects. In this work, we study the stability of higher-order two-term fractional linear…

The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…

We consider non linear elliptic equations of the form $\Delta u = f(u,\nabla u)$ for suitable analytic nonlinearity $f$, in the vinicity of infinity in $\mathbb{R}^d$, that is on the complement of a compact set.We show that there is a…

We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni-…

In this article, we show that a technique for showing well-posedness results for evolutionary equations in the sense of [13] established in [16] applies to a broader class of non-autonomous integro-differential-algebraic equations. Using…

We consider the maximal regularity problem for non-autonomous evolution equations of the form $u(t) + A(t) u(t) = f(t)$ with initial data $u(0) = u\_0$ . Each operator $A(t)$ is associated with a sesquilinear form $a(t; *, *)$ on a Hilbert…

We study homogenization problem for non-autonomous parabolic equations of the form $\partial_t u=L(t)u$ with an integral convolution type operator $L(t)$ that has a non-symmetric jump kernel which is periodic in spatial variables and in…

We prove the existence of solutions for an evolution quasi-variational inequality with a first order quasilinear operator and a variable convex set, which is characterized by a constraint on the absolute value of the gradient that depends…

We bound the difference between solutions $u$ and $v$ of $u_t = a\Delta u+\Div_x f+h$ and $v_t = b\Delta v+\Div_x g+k$ with initial data $\phi$ and $ \psi$, respectively, by $\Vert u(t,\cdot)-v(t,\cdot)\Vert_{L^p(E)}\le A_E(t)\Vert…

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u &+ \text{P.V.}\int_{\mathbb{R}^N}…

We deal with a planar differential system of the form \begin{equation*} \begin{cases} \, u' = h(t,v), \\ \, v' = - \lambda a(t) g(u), \end{cases} \end{equation*} where $h$ is $T$-periodic in the first variable and strictly increasing in the…

In the study of ordinary differential equations (ODEs) of the form $\hat{L}[y(x)]=f(x)$, where $\hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)\propto u(x)$ and $\hat{L}[u(x)]=0$, and…